Consider two circles $C_1$ and $C_2$, tangent at a point $p$, with
$C_2$ contained in the disk bounded by $C_1$. In the region between $C_1$ and $C_2$,
construct an infinite chain of circles $S_1$, $S_2$, $S_3$,…, each of them tangent to both
$C_1$ and $C_2$, and with $S_n$ tangent to $S_{n+1}$ at a point $p_n$, as seen in the figure
Now the strange thing thing about this question is that it is an exercise coming from an example sheet about conformal maps in the complex plane and Möbius transformations. Using geometry I am unable to solve this question, but since it is asked in a complex analysis course it suggests that the problem should be solved using complex analysis techniques. Unfortunately, I don't quite see the link between this question and complex analysis. Any help would be welcome!
Best Answer
Möbius transformations preserve angles and map circles to circles or lines. Here it is convenient to map the entire configuration with $f(z) = 1/(z-p)$. $f(p) = \infty$ (on the extended complex plane) so that the circles are mapped to parallel lines, and the region between them is mapped to a parallel strip.
If the circles are symmetric with respect to the real axis then their images are lines parallel to the y-axis.
Finding a chain of circles tangent to both lines and tangent to each other is not difficult. Then map the configuration back with $f^{-1}$.
Remark: Circles and lines in the complex plane are sometimes called “generalized circles” because they correspond exactly to circles on the Riemann sphere via the stereographic projection. With that notation one can simply say that Möbius transformations map generalized circles to generalized circles.